The Kain–Fritsch (KF) convective parameterization scheme (CPS) is an advanced version of the Fritsch-Chappell (FC) scheme, designed to better capture the complex dynamics of mesoscale convective systems. The KF scheme retains the fundamental assumption that convective effects remove convective available potential energy within a grid element over an advective time period but introduces significant improvements to more accurately represent physical processes not included in the FC scheme. These improvements include a new cloud model that more realistically distributes detrainment effects vertically, and the implementation of conservation principles for mass, thermal energy, total moisture, and momentum, which are crucial for larger-scale and longer-duration simulations. The KF scheme's mathematical formulation is detailed, focusing on the heating tendency due to subgrid-scale convective processes. It accounts for the contributions from updraft and downdraft mass fluxes, as well as the compensating mass flux in the surrounding environment. The scheme ensures that the net mass flux into a model layer is zero, balancing any surplus or deficit created by entrainment and detrainment. This indirect representation of updrafts and downdrafts allows the numerical model to accurately simulate convective processes without directly coupling the grid-scale temperature to the updraft or downdraft temperature unless there is active detrainment. Additionally, the KF scheme includes expressions for the net tendency of specific humidity due to subgrid-scale convection and the representation of liquid water detrainment from convective clouds. These enhancements make the KF scheme a more robust and realistic tool for modeling mesoscale convective systems.The Kain–Fritsch (KF) convective parameterization scheme (CPS) is an advanced version of the Fritsch-Chappell (FC) scheme, designed to better capture the complex dynamics of mesoscale convective systems. The KF scheme retains the fundamental assumption that convective effects remove convective available potential energy within a grid element over an advective time period but introduces significant improvements to more accurately represent physical processes not included in the FC scheme. These improvements include a new cloud model that more realistically distributes detrainment effects vertically, and the implementation of conservation principles for mass, thermal energy, total moisture, and momentum, which are crucial for larger-scale and longer-duration simulations. The KF scheme's mathematical formulation is detailed, focusing on the heating tendency due to subgrid-scale convective processes. It accounts for the contributions from updraft and downdraft mass fluxes, as well as the compensating mass flux in the surrounding environment. The scheme ensures that the net mass flux into a model layer is zero, balancing any surplus or deficit created by entrainment and detrainment. This indirect representation of updrafts and downdrafts allows the numerical model to accurately simulate convective processes without directly coupling the grid-scale temperature to the updraft or downdraft temperature unless there is active detrainment. Additionally, the KF scheme includes expressions for the net tendency of specific humidity due to subgrid-scale convection and the representation of liquid water detrainment from convective clouds. These enhancements make the KF scheme a more robust and realistic tool for modeling mesoscale convective systems.